Aspects of Finite Gaudin Models: Separation of Variables and Description from 3dBF Theory

In this thesis, we provide a separation of variables for quantum Gaudin models associated to low rank matrix Lie algebras, and a description of the finite classical Gaudin model for any semisimple Lie algebra from a gauge theory. We separate the variables for the quantum sl2-Gaudin model with irregular singularities, producing an explicit coordinate change and comparing this to the known solution provided by the Bethe Ansatz. We also produce a separation of variables of the gl3-Gaudin model with irregular singularities following previous work separating the variables in the XXX-chain. We do this both by directly taking the limit of the XXX case, and by working only in the Gaudin setting. The affine Gaudin model, associated with an untwisted affine Kac-Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We show that the finite classical Gaudin model, associated with a finite dimensional semisimple Lie algebra, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory with certain line defects in the Hamiltonian framework.